N2011 P2 Q11 [Modified]

A committee of \(10\) people is chosen at random from a group consisting of \(18\) women and \(12\) men. The number of women on the committee is denoted by \(R\).

1. Find the probability that \(R = 4\).
2. The most probable number of women on the committee is denoted by \(r\). By using the fact that \( \mathrm{P}(R = r) > \mathrm{P}(R = r + 1)\), show that \(r\) satisfies the inequality \[(r + 1)! (17 - r)! (9 - r)! (r + 3)! > r! (18 - r)! (10 - r)! (r + 2)!\] and use this inequality to find the value of \(r\).
3. Use the calculator to find \(\mathrm{E}(R)\) and \(\mathrm{Var}(R)\).